Subgroups — C₃×#10617372 #14415354

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4133	C₂	1	2	yes	maximal, complemented
2	C₃×#66059	C₂	1	1	yes	maximal, characteristic
3	C₃×#65787	C₂	1	2	yes	maximal, complemented
4	C₃×#65704	C₂	1	8	yes	maximal, complemented
5	C₃×#65682	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#65676	C₂	1	1	yes	maximal, characteristic, complemented
7	#10617372	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×#713	C₂²	1	2	yes	—
9	C₃×#4133	—	2	8	no	—
10	C₂×C₃×#700	C₂²	1	2	yes	—
11	C₂×C₃×#700	C₂²	1	4	yes	—
12	C₂×C₃×D₅⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×D₅⋊C₈	C₂²	1	4	yes	—
14	C₂×C₃×D₅⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
16	C₃×#4093	C₄	1	2	yes	—
17	C₃×#4041	C₄	1	4	yes	—
18	C₃×#4039	C₂²	1	4	yes	—
19	C₃×#4020	C₄	1	1	yes	characteristic
20	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
21	C₃×#4004	C₂²	1	4	yes	—
22	C₃×#4003	C₂²	1	8	yes	—
23	C₃×#4002	C₂²	1	4	yes	—
24	C₃×#2932	—	5	1	no	maximal, Hall
25	C₂×#4133	C₂×C₃	1	2	yes	complemented
26	#66059	C₂×C₃	1	1	yes	characteristic
27	#65787	C₂×C₃	1	2	yes	complemented
28	#65704	C₂×C₃	1	8	yes	complemented
29	#65682	C₂×C₃	1	1	yes	characteristic, complemented
30	#65676	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
32	C₃×#713	—	2	4	no	—
33	C₃×#713	—	2	4	no	—
34	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
35	C₂×C₃×D₂₀	C₂×C₄	1	4	yes	—
36	C₂×C₃×D₂₀	C₂×C₄	1	4	yes	—
37	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₄×D₅	C₂³	1	4	yes	—
39	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
40	C₂×C₃×#170	C₂³	1	4	yes	—
41	C₂×C₃×#170	C₂³	1	2	yes	—
42	C₂×C₃×#170	C₂³	1	2	yes	—
43	C₃×#700	—	2	4	no	—
44	C₃×#700	—	2	8	no	—
45	C₃×D₅⋊C₈	—	2	2	no	—
46	C₃×D₅⋊C₈	—	2	8	no	—
47	C₃×D₅⋊C₈	D₄	1	4	yes	complemented
48	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
49	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
50	C₃×#685	C₂×C₄	1	8	yes	—
51	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
52	C₃×C₄×Dic₅	C₂³	1	2	yes	—
53	C₂×C₃×#131	—	5	2	no	—
54	C₃×#532	—	5	1	no	—
55	C₃×#435	—	5	8	no	—
56	C₃×#423	—	5	1	no	—
57	C₃×#408	—	5	2	no	—
58	C₂×C₃×#97	—	5	1	no	—
59	C₂×#713	C₂²×C₃	1	2	yes	—
60	#4133	—	2	8	no	—
61	C₂×#700	C₂²×C₃	1	2	yes	—
62	C₂×#700	C₂²×C₃	1	4	yes	—
63	C₂×D₅⋊C₈	C₂²×C₃	1	1	yes	characteristic
64	C₂×D₅⋊C₈	C₂²×C₃	1	4	yes	—
65	C₂×D₅⋊C₈	C₂²×C₃	1	1	yes	characteristic
66	C₅×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
67	#4093	C₃×C₄	1	2	yes	—
68	#4041	C₃×C₄	1	4	yes	—
69	#4039	C₂²×C₃	1	4	yes	—
70	#4020	C₃×C₄	1	1	yes	characteristic
71	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
72	#4004	C₂²×C₃	1	4	yes	—
73	#4003	C₂²×C₃	1	8	yes	—
74	#4002	C₂²×C₃	1	4	yes	—
75	#2932	—	5	1	no	Sylow
76	C₂²×C₃×D₅	C₂²×C₄	1	4	yes	—
77	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
78	C₃×C₅×Q₈	—	2	4	no	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
81	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
83	C₂×C₃×Dic₅	C₂⁴	1	1	yes	characteristic
84	C₃×D₂₀	—	2	8	no	—
85	C₃×D₂₀	—	2	4	no	—
86	C₃×D₂₀	—	4	4	no	—
87	C₃×C₄×D₅	—	2	4	no	—
88	C₃×C₄×D₅	—	2	4	no	—
89	C₃×C₄×D₅	—	4	2	no	—
90	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
91	C₃×C₄×D₅	#41	1	2	yes	—
92	C₃×#170	—	2	4	no	—
93	C₃×#170	—	2	2	no	—
94	C₃×#170	C₂×D₄	1	4	yes	—
95	C₂×C₃×#41	—	5	2	no	—
96	C₃×#131	—	10	8	no	—
97	C₂×C₃×M₁₆	—	5	2	no	—
98	C₂×C₃×M₁₆	—	5	4	no	—
99	C₂²×C₃×C₈	—	5	1	no	—
100	C₂²×C₃×C₈	—	5	4	no	—
101	C₂²×C₃×C₈	—	5	1	no	—
102	C₃×C₄⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
103	C₃×#127	—	5	1	no	—
104	C₃×#124	—	5	2	no	—
105	C₃×#121	—	5	4	no	—
106	C₃×C₄×D₄	—	5	4	no	—
107	C₂×C₃×C₄²	—	5	1	no	—
108	C₃×C₄⋊C₈	—	5	4	no	—
109	C₃×C₂²⋊C₈	—	5	8	no	—
110	C₃×#97	—	5	4	no	—
111	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
112	#713	—	2	4	no	—
113	#713	—	2	4	no	—
114	C₂×D₂₀	C₂×C₃×C₄	1	2	yes	—
115	C₂×D₂₀	C₂×C₃×C₄	1	4	yes	—
116	C₂×D₂₀	C₂×C₃×C₄	1	4	yes	—
117	C₂×C₄×D₅	C₂×C₃×C₄	1	2	yes	—
118	C₂×C₄×D₅	C₂³×C₃	1	4	yes	—
119	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
120	C₂×#170	C₂³×C₃	1	4	yes	—
121	C₂×#170	C₂³×C₃	1	2	yes	—
122	C₂×#170	C₂³×C₃	1	2	yes	—
123	#700	—	2	4	no	—
124	#700	—	2	8	no	—
125	D₅⋊C₈	—	2	2	no	—
126	D₅⋊C₈	—	2	8	no	—
127	D₅⋊C₈	C₃×D₄	1	4	yes	complemented
128	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
129	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
130	#685	C₂×C₃×C₄	1	8	yes	—
131	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
132	C₄×Dic₅	C₂³×C₃	1	2	yes	—
133	C₂×#131	—	5	2	no	—
134	#532	—	5	1	no	—
135	#435	—	5	8	no	—
136	#423	—	5	1	no	—
137	#408	—	5	2	no	—
138	C₂×#97	—	5	1	no	—
139	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₃×D₅	—	2	2	no	—
141	C₂×C₃×D₅	—	2	4	no	—
142	C₂×C₃×D₅	—	2	8	no	—
143	C₂×C₃×D₅	C₄×D₄	1	2	yes	—
144	C₃×C₄×C₅	—	2	4	no	—
145	C₃×C₄×C₅	—	2	2	no	—
146	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
147	C₃×Dic₅	—	2	2	no	—
148	C₃×Dic₅	C₂²×D₄	1	1	yes	characteristic
149	C₃×Dic₅	C₂×#41	1	1	yes	characteristic
150	C₃×#41	—	10	4	no	—
151	C₃×#41	—	10	4	no	—
152	C₂×C₃×Q₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
153	C₂×C₃×D₄	—	5	4	no	—
154	C₂×C₃×D₄	—	5	2	no	—
155	C₂×C₃×D₄	—	5	4	no	—
156	C₂²×C₃×C₄	—	5	1	no	—
157	C₂²×C₃×C₄	—	5	4	no	—
158	C₂²×C₃×C₄	—	5	2	no	—
159	C₃×M₁₆	—	10	8	no	—
160	C₃×M₁₆	—	10	4	no	—
161	C₂×C₃×C₈	—	5	2	no	—
162	C₂×C₃×C₈	—	5	4	no	—
163	C₂×C₃×C₈	—	5	2	no	—
164	C₂×C₃×C₈	—	5	4	no	—
165	C₂×C₃×C₈	—	10	8	no	—
166	C₂×C₃×C₈	—	10	2	no	—
167	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	4	yes	—
168	C₃×C₂²⋊C₄	—	5	8	no	—
169	C₃×C₄²	—	5	1	no	—
170	C₃×C₄²	—	5	2	no	—
171	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
172	C₂²×D₅	C₂²×C₃×C₄	1	4	yes	—
173	C₂²×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
174	C₅×Q₈	—	2	4	no	—
175	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
176	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
177	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₂×Dic₅	C₂²×C₃×C₄	1	2	yes	—
179	C₂×Dic₅	C₂⁴×C₃	1	1	yes	characteristic
180	D₂₀	—	2	8	no	—
181	D₂₀	—	2	4	no	—
182	D₂₀	—	4	4	no	—
183	C₄×D₅	—	2	4	no	—
184	C₄×D₅	—	2	4	no	—
185	C₄×D₅	—	4	2	no	—
186	C₄×D₅	C₂×C₃×D₄	1	2	yes	—
187	C₄×D₅	C₃×#41	1	2	yes	—
188	#170	—	2	4	no	—
189	#170	—	2	2	no	—
190	#170	C₂×C₃×D₄	1	4	yes	—
191	C₂×#41	—	5	2	no	—
192	#131	—	10	8	no	—
193	C₂×M₁₆	—	5	2	no	—
194	C₂×M₁₆	—	5	4	no	—
195	C₂²×C₈	—	5	1	no	—
196	C₂²×C₈	—	5	4	no	—
197	C₂²×C₈	—	5	1	no	—
198	C₄⋊Q₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
199	#127	—	5	1	no	—
200	#124	—	5	2	no	—
201	#121	—	5	4	no	—
202	C₄×D₄	—	5	4	no	—
203	C₂×C₄²	—	5	1	no	—
204	C₄⋊C₈	—	5	4	no	—
205	C₂²⋊C₈	—	5	8	no	—
206	#97	—	5	4	no	—
207	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
208	C₂×C₃×C₅	#568	1	2	yes	—
209	C₃×D₅	—	2	2	no	—
210	C₃×D₅	—	4	4	no	—
211	C₂³×C₃	—	5	1	no	—
212	C₂³×C₃	—	5	4	no	—
213	C₃×Q₈	—	2	4	no	—
214	C₃×D₄	—	10	8	no	—
215	C₃×D₄	—	10	4	no	—
216	C₃×D₄	—	20	4	no	—
217	C₂×C₃×C₄	—	5	1	no	—
218	C₂×C₃×C₄	—	5	2	no	—
219	C₂×C₃×C₄	—	5	2	no	—
220	C₂×C₃×C₄	—	5	2	no	—
221	C₂×C₃×C₄	—	10	4	no	—
222	C₂×C₃×C₄	—	10	4	no	—
223	C₂×C₃×C₄	—	20	2	no	—
224	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
225	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	4	yes	—
226	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
227	C₃×C₈	—	5	4	no	—
228	C₃×C₈	—	10	2	no	—
229	C₃×C₈	—	10	4	no	—
230	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
231	C₂×D₅	—	2	2	no	—
232	C₂×D₅	—	2	8	no	—
233	C₂×D₅	—	2	4	no	—
234	C₂×D₅	C₃×C₄×D₄	1	2	yes	—
235	C₄×C₅	—	2	2	no	—
236	C₄×C₅	—	2	4	no	—
237	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
238	Dic₅	—	2	2	no	—
239	Dic₅	C₂²×C₃×D₄	1	1	yes	characteristic
240	Dic₅	C₂×C₃×#41	1	1	yes	characteristic
241	#41	—	10	4	no	—
242	#41	—	10	4	no	—
243	C₂×Q₈	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
244	C₂×D₄	—	5	4	no	—
245	C₂×D₄	—	5	2	no	—
246	C₂×D₄	—	5	4	no	—
247	C₂²×C₄	—	5	1	no	—
248	C₂²×C₄	—	5	4	no	—
249	C₂²×C₄	—	5	2	no	—
250	M₁₆	—	10	8	no	—
251	M₁₆	—	10	4	no	—
252	C₂×C₈	—	5	2	no	—
253	C₂×C₈	—	5	4	no	—
254	C₂×C₈	—	5	2	no	—
255	C₂×C₈	—	5	4	no	—
256	C₂×C₈	—	10	8	no	—
257	C₂×C₈	—	10	2	no	—
258	C₄⋊C₄	C₂×C₃×AGL(1,𝔽₅)	1	4	yes	—
259	C₂²⋊C₄	—	5	8	no	—
260	C₄²	—	5	2	no	—
261	C₄²	—	5	1	no	—
262	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
263	C₃×C₅	#2932	1	1	yes	characteristic, complemented, Hall
264	C₂²×C₃	—	5	2	no	—
265	C₂²×C₃	—	10	2	no	—
266	C₂²×C₃	—	10	4	no	—
267	C₂²×C₃	—	10	8	no	—
268	C₂²×C₃	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
269	C₃×C₄	—	2	4	no	—
270	C₃×C₄	—	2	2	no	—
271	C₃×C₄	—	5	1	no	—
272	C₃×C₄	—	5	1	no	—
273	C₃×C₄	—	10	2	no	—
274	C₃×C₄	D₄×AGL(1,𝔽₅)	1	2	yes	—
275	C₂×C₅	C₂×C₃×C₄×D₄	1	1	yes	characteristic
276	C₂×C₅	C₃×#568	1	2	yes	—
277	D₅	—	2	2	no	—
278	D₅	—	4	4	no	—
279	C₂³	—	5	1	no	—
280	C₂³	—	5	4	no	—
281	Q₈	—	2	4	no	—
282	D₄	—	10	8	no	—
283	D₄	—	10	4	no	—
284	D₄	—	20	4	no	—
285	C₂×C₄	—	5	1	no	—
286	C₂×C₄	—	5	2	no	—
287	C₂×C₄	—	5	2	no	—
288	C₂×C₄	—	5	2	no	—
289	C₂×C₄	—	10	4	no	—
290	C₂×C₄	—	10	4	no	—
291	C₂×C₄	—	20	2	no	—
292	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	2	yes	—
293	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	4	yes	—
294	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
295	C₈	—	5	4	no	—
296	C₈	—	10	4	no	—
297	C₈	—	10	2	no	—
298	C₂×C₃	—	10	2	no	—
299	C₂×C₃	—	20	4	no	—
300	C₂×C₃	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
301	C₂×C₃	#66260	1	2	yes	—
302	C₅	C₃×#2932	1	1	yes	characteristic, complemented, Sylow
303	C₂²	—	5	2	no	—
304	C₂²	—	10	2	no	—
305	C₂²	—	10	4	no	—
306	C₂²	—	10	8	no	—
307	C₂²	C₂³×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
308	C₄	—	2	4	no	—
309	C₄	—	2	2	no	—
310	C₄	—	5	1	no	—
311	C₄	—	5	1	no	—
312	C₄	—	10	2	no	—
313	C₄	C₃×D₄×AGL(1,𝔽₅)	1	2	yes	—
314	C₃	#10617372	1	1	yes	characteristic, direct product factor, Sylow
315	C₂	—	10	2	no	—
316	C₂	—	20	4	no	—
317	C₂	C₂×C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
318	C₂	C₃×#66260	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂³×C₃×C₄
Frattini	C₂²	C₂³×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×C₄⋊Q₈	C₄
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₃×#66059	C₂
cosocle	C₂×Dic₅	C₂⁴×C₃
hypercenter	C₃×C₄⋊Q₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2932
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄⋊Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₃×#4133	C₂	semidirect
3. C₃×#65787	C₂	semidirect
4. C₃×#65704	C₂	semidirect
5. C₃×#65682	C₂	semidirect
6. C₃×#65676	C₂	semidirect
7. #10617372	C₃	direct (N × H)
25. C₂×#4133	C₂×C₃	semidirect
27. #65787	C₂×C₃	semidirect
28. #65704	C₂×C₃	semidirect
29. #65682	C₂×C₃	semidirect
30. #65676	C₂×C₃	semidirect
47. C₃×D₅⋊C₈	D₄	semidirect
127. D₅⋊C₈	C₃×D₄	semidirect
263. C₃×C₅	#2932	semidirect
302. C₅	C₃×#2932	semidirect
314. C₃	#10617372	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×#4133	2
C₃×#66059	2
C₃×#65787	2
C₃×#65704	2
C₃×#65682	2
C₃×#65676	2
#10617372	3
C₃×#2932	5

Subgroups — C3×#10617372 #14415354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10617372 #14415354